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dc.contributor.authorMitchell, James David
dc.contributor.authorMorayne, Michal
dc.contributor.authorPeresse, Yann Hamon
dc.contributor.authorQuick, Martyn
dc.identifier.citationMitchell , J D , Morayne , M , Peresse , Y H & Quick , M 2010 , ' Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances ' , Annals of Pure and Applied Logic , vol. 161 , no. 12 , pp. 1471-1485 .
dc.identifier.otherPURE: 4157941
dc.identifier.otherPURE UUID: 6d3e27e5-0ec5-4fd5-a87e-7dfc3804c190
dc.identifier.otherScopus: 77955511177
dc.identifier.otherWOS: 000281923900003
dc.identifier.otherORCID: /0000-0002-5227-2994/work/58054906
dc.identifier.otherORCID: /0000-0002-5489-1617/work/73700779
dc.description.abstractLet ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0.
dc.relation.ispartofAnnals of Pure and Applied Logicen
dc.rightsThis is an author version of of this article. The published version Copyright © 2010 Elsevier B.V. is available at http://www.sciencedirect.comen
dc.subjectQA Mathematicsen
dc.titleGenerating transformation semigroups using endomorphisms of preorders, graphs, and tolerancesen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews.School of Mathematics and Statisticsen
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews.Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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